Environmental Engineering Reference
In-Depth Information
1.2.3
Statistical Weight of a State and Distributions of Particles in Gases
Above we used subscript i to refer to one particle state, whereas below we consider
a general case where i characterizes a set of degenerate states. We introduce the
statistical weight g i of a state that is a number of degenerate states i . For example,
a diatomic molecule in a rotational state with the rotational quantum number J
has the statistical weight g i
1, equal to the number of momentum projec-
tions on the molecular axis. Including accounting for the statistical weight, formula
(1.41) takes the form
D
2 J
C
Cg j exp
T ,
ε
j
n j
D
(1.42)
where C is the normalization factor and the subscript j refers to a group of states.
In particular, this formula gives the relation between the number densities N 0 and
N j of particles in the ground and excited states, respectively:
g 0 exp
T ,
N 0 g j
ε
j
N j
D
(1.43)
where
j is the excitation energy and g 0 and g j are the statistical weights of the
ground and excited states.
We now determine the statistical weight of states in a continuous spectrum. The
wave function of a free particle with momentum p x moving along the x -axis is
given by exp( ip x x /
ε
) to within an arbitrary factor if the particle is moving in the
positive direction and by exp(
ip x x /
) if the particle is moving in the negative
10 27 erg s is the Planck constant h divided
direction. (The quantity
„D
1.054
by 2
.) Suppose that the particle is in a potential well with infinitely high walls.
Theparticlecanmovefreelyintheregion0
π
L and the wave function at the
walls goes to zero. To construct a wave function that corresponds to free motion
inside the well and goes to zero at the walls, we superpose the basic free-particle
solutions, so
<
x
<
ψ D
C 1 exp( ip x x /
)
C
C 2 exp(
ip x x /
). From the boundary condi-
tion
ψ
(0)
D
0itfollowsthat
ψ D
C sin( p x x /
), and from the second boundary
condition
n ,where n is an integer. This procedure
thus yields the allowed quantum energies for a particle moving in a rectangular
well with infinitely high walls.
From this it follows that the number of states for a particle with momentum in
the range from p x to p x
ψ
( L )
D
0weobtain p x L /
„D π
),wherewetakeinto
account the two directions of the particle momentum. For a spatial interval dx ,the
number of particle states is
C
dp x is given by dn
D
Ldp x /(2
π
dp x dx
2
dn
D
.
π
Generalizing to the three-dimensional case, we obtain
dp x dx
2
dp y dy
2
dp z dz
2
d p d r
(2
dn
D
D
) 3 .
(1.44)
π
π
π
π
 
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